Outer Product: Difference between revisions

{{Built-in|Outer Product|<nowiki>∘.</nowiki>}}, or '''Table''' is a [[monadic operator]], which will produce a [[dyadic function]] when applied with a [[dyadic function]]. Outer product applies the [[operand]] function on each [[element]] of the left array with each element of the right array. It can be described as a shortcut for constructing nested [[wikipedia:for loop|for loop]]s, and as a structured [[wikipedia:Cartesian product|Cartesian product]] generalized to operations other than [[catenate|catenation]].

[[J]]'s Table differs from APL's Outer Product by enabling Cartesian pairing between [[cell]]s of [[rank]]s other than 0, and between cells of different rank for the left vs. right arguments, without needing to first [[enclose]] the relevant cells. It is defined as <syntaxhighlight lang=j inline>u"(lu, _)</syntaxhighlight>, where <syntaxhighlight lang=j inline>lu</syntaxhighlight> is the left rank of operand <syntaxhighlight lang=j inline>u</syntaxhighlight> (<syntaxhighlight lang=j inline>"</syntaxhighlight> and <syntaxhighlight lang=j inline>_</syntaxhighlight> denote [[rank_(operator)|Rank]] and infinity, respectively, in J).  

In APL this behavior can be accomplished by enclosing the relevant cells of each argument before applying the outer product operation (or by successive applications of Rank, e.g. <syntaxhighlight lang=apl inline>f⍤99 1⍤2 99</syntaxhighlight>). Conversely, the behavior of APL's Outer Product on nested inputs can be accomplished in terms of J's Table by adding the Each modifier, as in <syntaxhighlight lang=j inline>u&.>/</syntaxhighlight>.

The results of the individual applications of u are collectively [[frame]]d by the catenation of the frames of the left and right arguments relative to their lu- and ru-cells respectively, where lu and ru are the dyadic ranks of the operand u.  

Table does not allow for creating a Cartesian pairing involving cells of the left argument specified with negative rank (except trivially, in cases in which Table's application has no effect, e.g. <syntaxhighlight lang=j inline>u"_ _1"_2 _/</syntaxhighlight>). This is because negative assigned rank in J is encoded as infinite rank; in J's terminology, negative rank is "internal rank" only.